On the distribution of equivalence classes of random symmetric p-adic matrices

We consider random symmetric matrices with independent entries distributed according to the Haar measure on $\mathbb{Z}_p$ for odd primes $p$ and derive the distribution of their canonical form with respect to several equivalence relations. We give a few examples of applications including an alternative proof for the result of Bhargava, Cremona, Fisher, Jones, and Keating on the probability that a random quadratic form over $\mathbb{Z}_p$ has a non-trivial zero.